1. Calculating Nonlinear Response Functions from Gaussian-Markovian Quantum Fokker-Planck Approach Yoshitaka Tanimura Department of Chemistry Kyoto University

2. In an environment ・ Fluctuation ・ Dissipation Chemical processes in condensed phase matched equilibrium Heat bath model

3. 3 Model Hamiltonian (vibrational modes) T 1 + T 2 relaxationT 2 * relaxation

4. In the path integral representation (for a harmonic bath) where is the Feynman-Vernon influence functional. dissipation fluctuation

5. QFP equation for Gaussian-Markovian noise bath If we assume we have (colored noise) Markovian like noise For Hamiltonian If temp. is high

6. Density matrix elements where Time derivative of each parts are

7. we have where Then

8. We may evaluate by repeating the differentiation, then where is the density matrix for the element

9. in the Wigner representation; For nth member (similar to Kubo‘s stochstic Liouv. Eq.) Quantum Liouvillian is (YT & Kubo, JPSJ 1989)

10. G-M quantum Fokker-Planck eq (LL+SL) Quantum Liouvillian YT & Wolynes JCP 1993 YT (review), JPSJ 2006 ↑ temperature term qx (LL) + q 2 x (SL)

11. Physica meaning of Hierachy elements Dashed line represents the system-bath interactions (0 th order is exact) Major roles to calc. correlation func. (correlated effects)

12. 12 Two-time correlation function (Raman spectroscopy) This can be rewritten in the Wigner representation as where A and X stand for the operators and

13. 13 2D Raman (YT & Mukamel, JCP 1993) two-time variables t 1, t 2 three-body correlation function: is similarly expressed as

14. 14 Example ： anharmonicity 1) Harmonic 2) Morse Quantum Fokker-Planck (white noise at high temperature)

15. 15 1D Raman 2body correlation func.

16. 16 2DRaman Three body corre.

17. Kaufman, Heo & Fleming, PRL88, 207402(2002). 2D Raman (CS 2 ) Saito and Ohmine, PRL 88, 207401 (2002). Okumura & Tanimura, JCP 107, 2267 (1997) ExperimentMD simulationAnalytical result Anharmonisity + Nonlinear Polarizability

18. 18 Noise correlation and SL interactions T 1 + T 2 relaxationT 2 * relaxation Hamiltonian

19. potential system vs. Energy-level system LL + SL interactions T 1 +T 2 T 2 * different!

20. Third-order 2D IR t 2 =0 photon echo signals?

21. Linear-Linear and Linear-Square (JPSJ review) Fast modulationSlow modulation only q x j (q+q 2 ) x j only q 2 x j

22. 2D IR signal of HF liquid (from MD) Hasegawa and Tanimura, submitted to JCP Similar to slow modulation case of LL+SL model

23. Low temp. corrections of GM QFP eq. Dissipation Similar to GM caseMatsubara freq. correct. terms High (Matsubara) frequencies terms are approximated by this Fluctuation (former case was )

24. Ishizaki and Tanimura, J. Phys. Soc. Jpn. 74, 3134 (2005). GM master eq. with low temp. correction terms

25. 25 Application to two-mode system Acetylacetonato Rhodiumdicarbonyl(I) (RDC) in chloroform solution Tokmakoff Group@MIT,

26. 26 “ ＋ ”corr “ － ” corr non corr Anharmonic oscillators coupled with one- or two-bath a single bath two baths

27. 27 Effects of noise correlation “ ＋ ”corr “ － ” corr non corr Ishizaki and Tanimura, J. Phys. Chem. A (2007)

28. Conclusions Quantum Fokker-Planck Eq. for GM noise colored noise, strong system-bath coupling, low temp. (w. low temp. correction terms) Variety of applications (multi-level system, etc) Multi-dimensional spectroscopy critical check for theory Review : Y. Tanimura, J. Phys. Soc. Jpn, 75, 082001 (2006).